Phases of Play:
1: Players decide who goes first, future games other player goes first.
2: Players each take a turn by placing their pieces on the board one at a time
3: The players move their pieces around the board attempting to create a 3 in a row
4: The player who creates a 3 in a row removes those pieces from the board and can’t be used anymore in the current game.

Winning: The player who is left with 1 of their pieces on the board wins.

It seems a decent place to start for the game, sure. A couple initial thoughts.

- Try starting with a smaller board. A smaller board encourages players to cross paths with one another more frequently, as well as not sprawl out and take a leisurely approach to eliminating their own pieces.

- Might pieces stack upon one another? If they did, what sort of in-game situations/benefits/penalties would this present?

- There's potential to have 4- or 5-in-a-row as well with these kinds of games. In typical "match three" video games such as Bejeweled the player receives a benefit for arranging this. Additionally, this would potentially allow players to eliminate all their pieces from the board and their supply (as well as prevent them from doing so, should they not be paying attention). How would this change your game if you considered implementing it?

- Try starting with a smaller board. A smaller board encourages players to cross paths with one another more frequently, as well as not sprawl out and take a leisurely approach to eliminating their own pieces.

And you'd need a lot fewer pieces, because 31+31 = 62 (unless you could stack or something)

Also, what would prevent me from just putting all my pieces down in groups of 3 to start with? This is very easy to prototype.. have you actually tried playing a game of it?

I think the game as described could easily get bogged down into a tic-tac-toe kind of tie with everyone blocking each other's moves. Mechanisms to prevent that might present themselves while noodling around with a prototype.

For example, maybe you can only place within a certain radius of your existing pieces. The player who goes first might be content to play separately, but the second player has an incentive to reach across the board and cause some problems.

Stacking might be a good way to liberate blocked/trapped pieces. Maybe every piece beyond the third that's eliminated at once can be stacked onto existing pieces. A stack-of-two can be removed with any adjacent friendly piece because they add up to three. A completely surrounded piece can even get rescued by making a stack of three. The ultimate move would then be to arrange a diamond of four stacks-of-two, then place one piece in the middle to eliminate 9 simultaneously... those six could be stacked elsewhere leading to a chain reaction that could conceivably remove every piece in play. (Note, if your opponent doesn't put their own piece in that center spot, you're obviously playing a 4-year-old).